On Representation of the Reeb Graph as a Sub-Complex of Manifold

The Reeb graph $\mathcal{R}(f)$ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R}$ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $\pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.Comment: 18 page

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

In this work we present a construction of correspondence between epimorphisms $\varphi \colon \pi_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto their framed cobordism classes. In consequence, for closed manifolds any such $\varphi$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups providing a new purely geometrical-topological proof of the solution of this problem for surface groups.Comment: 23 page

Bourgin-Yang versions of the Borsuk-Ulam theorem for pp-toral groups

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $\dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$.Comment: Major revisio